Zhu et al. Fixed Point Theory and Applications 2012, 2012:159 http://www.fixedpointtheoryandapplications.com/content/2012/1/159
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Common fixed point theorems for fuzzy mappings in G-metric spaces Li Zhu1,2 , Chuan-Xi Zhu1* and Chun-Fang Chen1 *
Correspondence:
[email protected] 1 Department of Mathematics, Nanchang University, Nanchang, 330031, P.R. China Full list of author information is available at the end of the article
Abstract In this paper, we introduce the concept of Hausdorff G-metric in the space of fuzzy sets induced by the metric dG and obtain some results on Hausdorff G-metric. We also prove common fixed point theorems for a family of fuzzy self-mappings in the space of fuzzy sets on a complete G-metric space. MSC: 47H10; 54H25 Keywords: fuzzy set; DG,∞ -convergent; DG,∞ -Cauchy; fuzzy self-mapping; fixed point
1 Introduction and preliminaries Fixed point theory is very important in mathematics and has applications in many fields. A number of authors established fixed point theorems for various mappings in different metric spaces. In , Mustafa and Sims [] introduced the G-metric space as a generalization of metric spaces. We now recall some definitions and results in G-metric spaces in []. Definition . Let X be a nonempty set, and let G : X × X × X → R+ be a function satisfying: (G) (G) (G) (G) (G)
G(x, y, z) = if x = y = z, < G(x, x, y) for all x, y ∈ X with x = y, G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X, with y = z, G(x, y, z) = G(x, z, y) = G(y, z, x) = · · · (symmetry in all three variables), G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair (X, G) is a G-metric space. Lemma . Every G-metric space (X, G) defines a metric space (X, dG ) by dG (x, y) = G(x, y, y) + G(x, x, y),
for all x, y ∈ X.
Definition . Let (X, G) be a G-metric space. The sequence {xn } in X is said to be (i) G-convergent to x if for any ε > , there exists x ∈ X and N ∈ N such that G(x, xn , xm ) < ε, for all n, m ≥ N . (ii) G-Cauchy if for any ε > , there exists N ∈ N such that G(xn , xm , xl ) < ε, for all n, m, l ≥ N . © 2012 Zhu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Lemma . Let (X, G) be a G-metric space, then for a sequence {xn } in X and point x ∈ X the following are equivalent: (i) {xn } is G-convergent to x. (ii) G(xn , xn , x) → as n → +∞. (iii) G(xn , x, x) → as n → +∞. (iv) G(xm , xn , x) → as m, n → +∞. Lemma . Let (X, G) be a G-metric space, then for a sequence {xn } in X, the following are equivalent: (i) The sequence {xn } is G-Cauchy. (ii) For any ε > , there exists N ∈ N such that G(xn , xm , xm ) < ε, for all n, m ≥ N . (iii) {xn } is a Cauchy sequence in the metric space (X, dG ). Definition . A G-metric space (X, G) is said to be G-complete if every G-Cauchy sequence in (X, G) is G-convergent in (X, G). Lemma . A G-metric space (X, G) is G-complete if and only if (X, dG ) is a complete metric space. Based on the notion of G-metric spaces, many authors obtained fixed point theorems for mappings satisfying different contractive-type conditions in G-metric spaces (see, e.g., [– ]) and in partially ordered G-metric spaces (see, e.g., [–]). Recently, Kaewcharoen and Kaewkhao [] introduced the following concepts. Let X be a G-metric space and CB(X) the family of all nonempty closed bounded subsets of X. Let H(·, ·, ·) be the Hausdorff G-distance on CB(X), i.e., HG (A, B, C) = max sup G(x, B, C), sup G(x, C, A), sup G(x, A, B) , x∈A
x∈B
x∈C
where G(x, B, C) = dG (x, B) + dG (B, C) + dG (x, C), dG (x, B) = inf dG (x, y), y∈B
dG (A, B) = inf dG (x, y). x∈A,y∈B
Kaewcharoen and Kaewkhao [] and Tahat et al. [] obtained some common fixed point theorems for single-valued and multi-valued mappings in G-metric spaces. The existence of fixed points of fuzzy mappings has been an active area of research interest since Heilpern [] introduced the concept of fuzzy mappings in . Many results have appeared related to fixed points for fuzzy mappings in ordinary metric spaces (see, e.g., [–]). Qiu and Shu [, ] proved some fixed point theorems for fuzzy selfmappings in ordinary metric spaces. However, there are very few results on fuzzy selfmappings in G-metric spaces. The purpose of this paper is to introduce the notion of Hausdorff G-metric in the space of fuzzy sets which extends the Hausdorff G-distance in []. We also establish common fixed point theorems for a family of fuzzy self-mappings in the space of fuzzy sets on a complete G-metric space.
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2 A Hausdorff G-metric in the space of fuzzy sets Let (X, dG ) be a metric space, a fuzzy set in X is a function with domain X and values in I = [, ]. If μ is a fuzzy set and x ∈ X, then the function value μ(x) is called the grade of membership of x in μ. The α-level set of μ, denoted by [μ]α , is defined as [μ]α = x : μ(x) ≥ α , [μ] = x : μ(x) > ,
if α ∈ (, ],
where B is the closure of the non-fuzzy set B. Let C(X) be the family of all nonempty compact subsets of X. Denote by C(X) the totality of fuzzy sets which satisfy that for each α ∈ I, [μ]α ∈ C(X). Let μ , μ ∈ C(X), then μ is said to be more accurate than μ , denoted by μ ⊂ μ , if and only if μ (x) ≤ μ (x) for each x ∈ X. μ = μ if and only if μ ⊂ μ and μ ⊂ μ . Let μ , μ ∈ C(X), define D∞ (μ , μ ) = sup H [μ ]α , [μ ]α ≤α≤
= sup max sup dG x, [μ ]α , sup dG y, [μ ]α . ≤α≤
x∈[μ ]α
y∈[μ ]α
Lemma . [] The metric space (C(X), D∞ ) is complete provided (X, dG ) is complete. For μ , μ , μ ∈ C(X), α ∈ I, we define: Gα (μ , μ , μ ) = G [μ ]α , [μ ]α , [μ ]α = sup G x, [μ ]α , [μ ]α , x∈[μ ]α
G∞ (μ , μ , μ ) = sup Gα (μ , μ , μ ), ≤α≤
DG,α (μ , μ , μ ) = HG [μ ]α , [μ ]α , [μ ]α = max sup G x, [μ ]α , [μ ]α , sup G x, [μ ]α , [μ ]α , sup G x, [μ ]α , [μ ]α x∈[μ ]α
x∈[μ ]α
= max Gα (μ , μ , μ ), Gα (μ , μ , μ ), Gα (μ , μ , μ ) ,
x∈[μ ]α
DG,∞ (μ , μ , μ ) = sup DG,α (μ , μ , μ ) ≤α≤
= sup max Gα (μ , μ , μ ), Gα (μ , μ , μ ), Gα (μ , μ , μ ) ≤α≤
= max sup Gα (μ , μ , μ ), sup Gα (μ , μ , μ ), sup Gα (μ , μ , μ ) ≤α≤
≤α≤
≤α≤
= max G∞ (μ , μ , μ ), G∞ (μ , μ , μ ), G∞ (μ , μ , μ ) . Proposition . If A, B ∈ C(X) and x ∈ A, then there exists y ∈ B such that G(x, y, y) + G(y, x, x) ≤ HG (A, B, B).
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Proof For x ∈ A, there exists y ∈ B such that dG (x, y) = dG (x, B) = G(x, B, B), it follows that G(x, y, y) + G(y, x, x) = G(x, B, B) ≤ HG (A, B, B).
Proposition . If A, B ∈ C(X) and A ⊆ A, then there exists B ∈ C(X) such that B ⊆ B and HG (A , B , B ) ≤ HG (A, B, B). Proof Let C = {y : there exists x ∈ A , such that (G(x, y, y) + G(y, x, x)) ≤ HG (A, B, B)} and let B = C ∩ B. For any x ∈ A ⊆ A and B ∈ C(X), Proposition . implies that B is nonempty. Moreover, for any x ∈ A , there exists y ∈ B such that [G(x, y, y) + G(y, x, x)] ≤ HG (A, B, B). It follows that G(A , B , B ) = sup G(x, B , B ) = sup dG (x, B ) x∈A
x∈A
= sup inf G(x, y, y) + G(y, x, x) ≤ HG (A, B, B).
()
x∈A y∈B
On the other hand, for any y ∈ B , there exists x ∈ A such that [G(x, y, y) + G(y, x, x)] ≤ HG (A, B, B). Hence, G(B , B , A ) = sup G(y, B , A ) = sup dG (y, A ) + sup dG (y, B ) + dG (A , B ) y∈B
y∈B
y∈B
= sup inf G(x, y, y) + G(y, x, x) + + y∈B x∈A
inf
x∈A ,y∈B
G(x, y, y) + G(y, x, x)
≤ sup inf G(x, y, y) + G(y, x, x) + sup inf G(x, y, y) + G(y, x, x) y∈B x∈A
y∈B x∈A
= sup inf G(x, y, y) + G(y, x, x) ≤ HG (A, B, B). y∈B x∈A
()
From () and (), we have HG (A , B , B ) ≤ HG (A, B, B). Finally, we can conclude that B ∈ C(X) from the closeness of C and the compactness of B. Proposition . Let μ , μ ∈ C(X) and μ ⊂ μ , then there exists μ ∈ C(X) such that μ ⊂ μ and DG,∞ (μ , μ , μ ) ≤ DG,∞ (μ , μ , μ ).
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Proof Let α ∈ I, by μ ⊂ μ , we have [μ ]α ⊆ [μ ]α . Let Cα = y : there exists x ∈ [μ ]α such that G(x, y, y) + G(y, x, x) ≤ DG,∞ (μ , μ , μ ) , Dα = z : dG z, [μ ]α ≤ DG,∞ (μ , μ , μ ) , we can get that Cα = Dα . Let Bα = Dα ∩ [μ ]α , then Bα is nonempty compact and Bα ⊆ Bβ , for ≤ β ≤ α ≤ . From the proof of Proposition ., we have HG [μ ]α , Bα , Bα ≤ DG,∞ (μ , μ , μ ). Similar to the proof of Theorem in [], we can conclude that there exists a fuzzy set μ such that [μ ]α = Bα for α ∈ I. By the compactness of Bα , we have μ ∈ C(X). Therefore, DG,∞ (μ , μ , μ ) ≤ DG,∞ (μ , μ , μ ).
Proposition . Let X be a nonempty set. For any μ , μ , μ ∈ C(X), the following properties hold: (i) DG,∞ (μ , μ , μ ) = if and only if μ = μ = μ , (ii) < DG,∞ (μ , μ , μ ) for all μ , μ ∈ C(X) with μ = μ , (iii) DG,∞ (μ , μ , μ ) ≤ DG,∞ (μ , μ , μ ) for all μ , μ , μ ∈ C(X) with μ = μ , (iv) DG,∞ (μ , μ , μ ) = DG,∞ (μ , μ , μ ) = DG,∞ (μ , μ , μ ) = · · · (symmetry in all three variables), (v) DG,∞ (μ , μ , μ ) ≤ DG,∞ (μ , μ, μ) + DG,∞ (μ, μ , μ ). Proof The properties (i), (ii) and (iv) are readily derived from the definition of DG,∞ . First, we prove the property (iii). For any α ∈ I and x ∈ [μ ]α , y ∈ [μ ]α and z ∈ [μ ]α , we have dG (x, y) – dG (x, z) – dG (z, y) ≤ , it follows that dG (x, y) – dG x, [μ ]α – dG [μ ]α , [μ ]α ≤ sup dG (x, y) – inf dG (x, z) – z∈[μ ]α
y∈[μ ]α
=
inf
y∈[μ ]α ,z∈[μ ]α
dG (z, y)
dG (x, y) – dG (x, z) – dG (z, y) ≤ .
sup y∈[μ ]α ,z∈[μ ]α
This implies that inf
x∈[μ ]α ,y∈[μ ]α
dG (x, y) – sup dG x, [μ ]α ≤ dG [μ ]α , [μ ]α . x∈[μ ]α
Then, dG [μ ]α , [μ ]α ≤ sup dG x, [μ ]α + dG [μ ]α , [μ ]α . x∈[μ ]α
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Hence, G [μ ]α , [μ ]α , [μ ]α = sup dG x, [μ ]α + dG [μ ]α , [μ ]α x∈[μ ]α
≤ G [μ ]α , [μ ]α , [μ ]α = sup dG x, [μ ]α + sup dG x, [μ ]α + dG [μ ]α , [μ ]α . x∈[μ ]α
()
x∈[μ ]α
Similarly, we can prove that G [μ ]α , [μ ]α , [μ ]α ≤ G [μ ]α , [μ ]α , [μ ]α .
()
By () and (), we have DG,∞ (μ , μ , μ ) = sup DG,α (μ , μ , μ ) α∈I
= sup max G [μ ]α , [μ ]α , [μ ]α , G [μ ]α , [μ ]α , [μ ]α α∈I
≤ DG,∞ (μ , μ , μ ) = sup DG,α (μ , μ , μ ) α∈I
= sup max G [μ ]α , [μ ]α , [μ ]α , G [μ ]α , [μ ]α , [μ ]α , G [μ ]α , [μ ]α , [μ ]α . α∈I
Now, we prove the property (v). For any α ∈ I and x ∈ [μ ]α , y ∈ [μ]α , we have dG x, [μ ]α ≤ dG (x, y) + dG y, [μ ]α , it follows that sup dG x, [μ ]α ≤ sup dG x, [μ]α + sup dG y, [μ ]α .
x∈[μ ]α
x∈[μ ]α
()
y∈[μ]α
From () and dG [μ ]α , [μ ]α ≤ dG [μ]α , [μ ]α + dG [μ]α , [μ ]α , we have Gα (μ , μ , μ ) = G [μ ]α , [μ ]α , [μ ]α = sup dG x, [μ ]α + dG x, [μ ]α + dG [μ ]α , [μ ]α x∈[μ ]α
≤ sup dG x, [μ]α + dG x, [μ ]α + dG [μ]α , [μ ]α x∈[μ ]α
+ sup dG y, [μ ]α + dG y, [μ]α + dG [μ]α , [μ ]α y∈[μ]α
= Gα (μ , μ , μ) + Gα (μ, μ, μ ).
()
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Similarly, we can obtain that Gα (μ , μ , μ ) ≤ Gα (μ , μ , μ) + Gα (μ, μ, μ ),
()
Gα (μ , μ , μ ) ≤ Gα (μ, μ , μ ) + Gα (μ , μ, μ).
()
By (), () and (), we have DG,∞ (μ , μ , μ ) = sup max Gα (μ , μ , μ ), Gα (μ , μ , μ ), Gα (μ , μ , μ ) ≤α≤
≤ sup max Gα (μ , μ , μ), Gα (μ , μ , μ), Gα (μ, μ , μ ) ≤α≤
+ sup max Gα (μ, μ, μ ), Gα (μ, μ, μ ), Gα (μ , μ, μ) ≤α≤
= DG,∞ (μ, μ , μ ) + DG,∞ (μ , μ, μ).
Remark . Proposition . implies that DG,∞ is a G-metric in C(X), or more specially a Hausdorff G-metric in C(X). Definition . Let (C(X), DG,∞ ) be a metric space. The sequence {μn } in C(X) is said to be (i) DG,∞ -convergent to μ if for every ε > , there exists μ ∈ C(X) and N ∈ N such that DG,∞ (μ, μn , μm ) < ε for all n, m ≥ N , (ii) DG,∞ -Cauchy if for every ε > , there exists N ∈ N such that DG,∞ (μn , μm , μl ) < ε for all n, m, l ≥ N . Proposition . Let (C(X), DG,∞ ) be a metric space, then for a sequence {μn } ⊂ C(X) and μ ∈ C(X), the following are equivalent: (i) {μn } is DG,∞ -convergent to μ. (ii) D∞ (μ, μn ) → as n → +∞. (iii) DG,∞ (μn , μn , μ) → as n → +∞. (iv) DG,∞ (μ, μ, μn ) → as n → +∞. Proof Since DG,∞ is a G-metric, Lemma . implies that (i), (iii) and (iv) are equivalent. Now, we prove that (ii) is also an equivalent condition. “(i) =⇒ (ii)” Suppose DG,∞ (μ, μn , μm ) → as n, m → +∞, then G∞ (μ, μn , μm ) = sup sup dG x, [μn ]α + dG x, [μm ]α + dG [μn ]α , [μm ]α → ≤α≤ x∈[μ]α
and G∞ (μn , μ, μm ) = sup
sup dG x, [μ]α + dG x, [μm ]α + dG [μ]α , [μm ]α → .
≤α≤ x∈[μn ]α
Thus, for any α ∈ I, sup dG x, [μn ]α → as n → +∞,
x∈[μ]α
()
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and sup dG x, [μ]α → as n → +∞.
()
x∈[μn ]α
It follows that D∞ (μ, μn ) = sup H [μ]α , [μn ]α → as n → +∞. ≤α≤
“(ii) =⇒ (i)” Suppose D∞ (μ, μn ) → as n → +∞, then () and () hold. Moreover, ≤ dG ([μn ]α , [μ]α ) ≤ supx∈[μn ]α [dG (x, [μ]α )] implies that as n → +∞, dG [μn ]α , [μ]α → .
()
From (), () and (), we have as n → +∞, DG,∞ (μn , μn , μ) = sup max G [μn ]α , [μn ]α , [μ]α , G [μ]α , [μn ]α , [μn ]α ≤α≤
= sup max sup dG x, [μ]α + dG [μn ]α , [μ]α , sup dG x, [μn ]α → . ≤α≤
x∈[μn ]α
x∈[μ]α
Thus, from ≤ DG,∞ (μ, μn , μm ) = DG,∞ (μn , μ, μm ) ≤ DG,∞ (μn , μ, μ) + DG,∞ (μ, μ, μm ) ≤ DG,∞ (μ, μn , μn ) + DG,∞ (μn , μ, μn ) + DG,∞ (μ, μm , μm ) + DG,∞ (μm , μ, μm ) = DG,∞ (μn , μn , μ) + DG,∞ (μm , μm , μ), we can conclude that DG,∞ (μ, μn , μm ) → as n, m → +∞.
Proposition . Let (C(X), DG,∞ ) be a metric space and {μn } a sequence in C(X), then the following are equivalent: (i) The sequence {μn } is DG,∞ -Cauchy. (ii) For every ε > , there exists N ∈ N such that DG,∞ (μn , μm , μm ) < ε for all n, m > N . (iii) {μn } is a Cauchy sequence in the metric space (C(X), D∞ ). Proof “(i) ⇐⇒ (ii)” is evidence. “(ii) =⇒ (iii)” Suppose that for every ε > , there exists N ∈ N such that DG,∞ (μn , μm , μm ) < ε, for all n, m > N , then as n, m → +∞, G∞ (μn , μm , μm ) →
()
G∞ (μm , μm , μn ) → .
()
and
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It follows that sup dG x, [μm ]α = G∞ (μn , μm , μm ) → . ≤α≤ x∈[μn ]α sup
()
From () and ≤ sup
sup dG x, [μn ]α ≤ G∞ (μm , μm , μn ),
≤α≤ x∈[μm ]α
we have sup
sup dG x, [μn ]α →
as n, m → +∞.
≤α≤ x∈[μm ]α
()
By () and (), we have D∞ (μn , μm ) → as n, m → +∞, that is, {μn } is a Cauchy sequence in the metric space (C(X), D∞ ). “(iii) =⇒ (ii)” Suppose D∞ (μn , μm ) → as n, m → +∞, then () and () hold. Moreover, ≤ sup dG [μm ]α , [μn ]α ≤ sup ≤α≤
sup dG x, [μn ]α
≤α≤ x∈[μm ]α
and () imply that sup dG [μm ]α , [μn ]α →
≤α≤
as n, m → +∞.
()
From (), () and (), we have as n, m → +∞, G∞ (μn , μm , μm ) = sup G [μn ]α , [μm ]α , [μm ]α ≤α≤
= sup
sup dG x, [μm ]α →
()
≤α≤ x∈[μn ]α
and G∞ (μm , μm , μn ) = sup G [μm ]α , [μm ]α , [μn ]α ≤α≤
= sup
sup
≤α≤ x∈[μm ]α
dG x, [μn ]α + dG [μm ]α , [μn ]α → .
()
We can get from () and () that DG,∞ (μn , μm , μm ) → as n, m → +∞.
The next proposition follows directly from Lemma ., Lemma ., Proposition . and Proposition ..
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Proposition . The metric space (C(X), DG,∞ ) is complete provided (X, G) is G-complete. From the definitions of G∞ and DG,∞ , we can get the next proposition readily. Proposition . If μ, μ , μ ∈ C(X) and μ ⊂ μ , then (i) G∞ (μ , μ, μ) ≤ G∞ (μ , μ, μ), (ii) G∞ (μ, μ , μ ) ≤ G∞ (μ, μ , μ ) ≤ DG,∞ (μ, μ , μ ), (iii) G∞ (μ , μ , μ ) = .
3 Fixed point theorems for fuzzy self-mappings In this section, we establish two fixed point theorems for fuzzy self-mappings. First, we recall the concept of a fuzzy self-mapping in []. Definition . [] Let X be a metric space. A mapping F is said to be a fuzzy self-mapping if and only if F is a mapping from the space C(X) into C(X), i.e., F(μ) ∈ C(X) for each μ ∈ C(X). μ ∈ C(X) is said to be a fixed point of a fuzzy self-mapping F of C(X) if and only if μ ⊂ F(μ ). Let denote all functions φ : [, +∞) → [, +∞) satisfying: (i) φ is non-decreasing and continuous from the right,
∞ n n (ii) n= φ (t) < +∞, for all t > , where φ denotes the nth iterative function of φ. Remark . It can be directly verified that for any φ ∈ and all t > , φ(t) < t. Theorem . Let (X, G) be a G-complete metric space and {Ti }∞ i= a sequence of fuzzy selfmappings of C(X). Suppose that for each μ , μ ∈ C(X) and for arbitrary positive integers i and j, i = j, DG,∞ (Ti μ , Tj μ , Tj μ ) ≤ φ max DG,∞ (μ , μ , μ ), G∞ (μ , Ti μ , Ti μ ), G∞ (μ , Tj μ , Tj μ ),
G∞ (μ , Tj μ , Tj μ ) + G∞ (μ , Ti μ , Ti μ ) ,
()
where φ ∈ . Then there exists at least one μ* ∈ C(X) such that μ* ⊂ Ti μ* for all i ∈ Z+ . Proof Let μ ∈ C(X) and μ ⊂ T μ , by Proposition ., there exists μ ∈ C(X) such that μ ⊂ T μ and DG,∞ (μ , μ , μ ) ≤ DG,∞ (T μ , T μ , T μ ). Again by Proposition ., we can find μ ∈ C(X) such that μ ⊂ T μ and DG,∞ (μ , μ , μ ) ≤ DG,∞ (T μ , T μ , T μ ). Continuing this process, we can construct a sequence {μn } in C(X) such that μn+ ⊂ Tn+ μn ,
n = , , , . . .
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and DG,∞ (μn , μn+ , μn+ ) ≤ DG,∞ (Tn μn– , Tn+ μn , Tn+ μn ),
n = , , . . . .
()
By (), (), Proposition . and (v) in Proposition ., we have DG,∞ (μn , μn+ , μn+ ) ≤ DG,∞ (Tn μn– , Tn+ μn , Tn+ μn ) ≤ φ max DG,∞ (μn– , μn , μn ), G∞ (μn– , Tn μn– , Tn μn– ), G∞ (μn , Tn+ μn , Tn+ μn ),
G∞ (μn– , Tn+ μn , Tn+ μn ) + G∞ (μn , Tn μn– , Tn μn– ) ≤ φ max DG,∞ (μn– , μn , μn ), DG,∞ (μn– , μn , μn ), DG,∞ (μn , μn+ , μn+ ),
DG,∞ (μn– , μn+ , μn+ ) + ≤ φ max DG,∞ (μn– , μn , μn ), DG,∞ (μn , μn+ , μn+ ), DG,∞ (μn– , μn , μn ) + DG,∞ (μn , μn+ , μn+ )
= φ(max{DG,∞ (μn– , μn , μn ), DG,∞ (μn , μn+ , μn+ ),
n = , , , . . . .
()
Suppose that ≤ DG,∞ (μn– , μn , μn ) < DG,∞ (μn , μn+ , μn+ ), then DG,∞ (μn , μn+ , μn+ ) ≤ φ DG,∞ (μn , μn+ , μn+ ) < DG,∞ (μn , μn+ , μn+ ), which is a contradiction since DG,∞ (μn , μn+ , μn+ ) > . Hence, DG,∞ (μn , μn+ , μn+ ) ≤ DG,∞ (μn– , μn , μn )
()
DG,∞ (μn , μn+ , μn+ ) ≤ φ DG,∞ (μn– , μn , μn ) ≤ · · · ≤ φ n DG,∞ (μ , μ , μ ) .
()
and
Now, we prove that {μn } is a DG,∞ -Cauchy sequence. For positive integers m, n, we distinguish the following two cases. Case . If m > n, then DG,∞ (μn , μm , μm ) ≤ DG,∞ (μn , μn+ , μn+ ) + DG,∞ (μn+ , μn+ , μn+ ) + · · · + DG,∞ (μm– , μm , μm ) =
m– i=n
DG,∞ (μi , μi+ , μi+ ) ≤
m– i=n
φ i DG,∞ (μ , μ , μ ) .
()
Zhu et al. Fixed Point Theory and Applications 2012, 2012:159 http://www.fixedpointtheoryandapplications.com/content/2012/1/159
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Assume that DG,∞ (μ , μ , μ ) = , then μ = μ . Inequality () implies that DG,∞ (μ , μ , μ ) ≤ φ max DG,∞ (μ , μ , μ ), DG,∞ (μ , μ , μ ) = φ DG,∞ (μ , μ , μ ) . It follows from φ(t) < t that DG,∞ (μ , μ , μ ) = , that is, μ = μ . By induction, we have μ = μ = · · · = μk = · · · . Thus, μ = μk ⊂ Tk μk– = Tk μ , k = , , . . . .
i Suppose that DG,∞ (μ , μ , μ ) > . ∞ i= φ (t) < ∞ and () yield that DG,∞ (μn , μm , μm ) → ,
as n, m → +∞.
()
Case . If m < n, from () and ≤ DG,∞ (μn , μm , μm ) = DG,∞ (μm , μm , μn ) ≤ DG,∞ (μm , μn , μn ) + DG,∞ (μn , μm , μn ) = DG,∞ (μm , μn , μn ), we can get that DG,∞ (μn , μm , μm ) → as n, m → +∞.
()
Thus, () and () imply that {μn } is a DG,∞ -Cauchy sequence. As (X, G) is G-complete, by Proposition ., we conclude that (C(X), DG,∞ ) is complete. There exists μ* ∈ C(X) such that DG,∞ (μ* , μm , μm ) → as m → +∞. Now, Proposition . and () imply that G∞ (μ* , Ti μ* , Ti μ* ) ≤ G∞ (μ* , μj , μj ) + G∞ (μj , Ti μ* , Ti μ* ) ≤ G∞ (μ* , μj , μj ) + G∞ (Tj μj– , Ti μ* , Ti μ* ) ≤ G∞ (μ* , μj , μj ) + DG,∞ (Tj μj– , Ti μ* , Ti μ* ) ≤ G∞ (μ* , μj , μj ) + φ max DG,∞ (μj– , μ* , μ* ), G∞ (μj– , Tj μj– , Tj μj– ), G∞ (μ* , Ti μ* , Ti μ* ),
G∞ (μj– , Ti μ* , Ti μ* ) + G∞ (μ* , Tj μj– , Tj μj– ) ≤ DG,∞ (μ* , μj , μj ) + φ max DG,∞ (μj– , μ* , μ* ), DG,∞ (μj– , μj , μj ), G∞ (μ* , Ti μ* , Ti μ* ), G∞ (μj– , μ* , μ* ) + G∞ (μ* , Ti μ* , Ti μ* ) + G∞ (μ* , μj , μj ) + G∞ (μj , Tj μj– , Tj μj– ) ≤ DG,∞ (μ* , μj , μj )
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+ φ max DG,∞ (μj– , μ* , μ* ), DG,∞ (μj– , μ* , μ* ) + DG,∞ (μ* , μj , μj ), G∞ (μ* , Ti μ* , Ti μ* ),
G∞ (μj– , μ* , μ* )
+ G∞ (μ* , Ti μ* , Ti μ* ) + G∞ (μ* , μj , μj ) +
.
()
Letting j → +∞, we can see from () and Proposition . that G∞ (μ* , Ti μ* , Ti μ* ) ≤ φ G∞ (μ* , Ti μ* , Ti μ* ) . It implies that G∞ (μ* , Ti μ* , Ti μ* ) = , that is, μ* ⊂ Ti μ* . If in Theorem . we choose φ(t) = kt, where k ∈ (, ) is a constant, we obtain the following corollary. Corollary . Let (X, G) be a G-complete metric space and {Ti }∞ i= a sequence of fuzzy selfmappings of C(X). Suppose that for each μ , μ ∈ C(X) and for arbitrary positive integers i and j, i = j, DG,∞ (Ti μ , Tj μ , Tj μ ) ≤ k max DG,∞ (μ , μ , μ ), G∞ (μ , Ti μ , Ti μ ), G∞ (μ , Tj μ , Tj μ ),
G∞ (μ , Tj μ , Tj μ ) + G∞ (μ , Ti μ , Ti μ ) , where k ∈ (, ). Then there exists at least one μ* ∈ C(X) such that μ* ⊂ Ti μ* for all i ∈ Z+ . The following example illustrates Theorem .. Example . Let X = {, , , , . . .}. Define G : X × X × X → X by ⎧ ⎪ x + y + z, if x, y, z are all distinct and different from zero; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x + z, if x = y = z and all are different from zero; ⎪ ⎪ ⎪ ⎪ ⎨y + z + , if x = , y = z and y, z are different from zero; G(x, y, z) = ⎪ ⎪y + , if x = , y = z = ; ⎪ ⎪ ⎪ ⎪ ⎪ z + , if x = , y = , z = ; ⎪ ⎪ ⎪ ⎪ ⎩, if x = y = z. Then X is a complete nonsymmetric G-metric space []. For μ, ν ∈ C(X), y ∈ X and λ > , owing to Zadeh’s extension principle [], scalar multiplication and addition are defined by y (λμ)(y) = μ λ and (μ + ν)(x) =
sup x ,x :x +x =x
min μ(x ), ν(x ) .
Zhu et al. Fixed Point Theory and Applications 2012, 2012:159 http://www.fixedpointtheoryandapplications.com/content/2012/1/159
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For any < a < and μ, ν, ω ∈ C(X), we can get easily from the definition of G(x, y, z) that DG,∞ (aμ, aν, aω) = aDG,∞ (μ, ν, ω)
()
DG,∞ (μ, aν, aν) ≤ DG,∞ (μ, ν, ν).
()
and
Now, suppose < p ≤ , define μ : X → C(X) by
μ (x) =
⎧ ⎨p,
if x = ;
⎩, otherwise.
Suppose < q < , define {Ti }∞ i= a sequence of fuzzy self-mappings of C(X) as Ti (μ) = qi μ + μ
for any μ ∈ C(X).
For any i, j ∈ Z+ , without loss of generality, suppose i < j. For each μ , μ ∈ C(X), by (), () and the definition of α-level set, we have DG,∞ (Ti μ , Tj μ , Tj μ ) = DG,∞ qi μ + μ , qj μ + μ , qj μ + μ ≤ qi DG,∞ μ , qj–i μ , qj–i μ ≤ qi DG,∞ (μ , μ , μ ) ≤ qi max DG,∞ (μ , μ , μ ), G∞ (μ , Ti μ , Ti μ ), G∞ (μ , Tj μ , Tj μ ),
G∞ (μ , Tj μ , Tj μ ) + G∞ (μ , Ti μ , Ti μ ) . i Therefore, {Ti }∞ i= satisfy the conditions of Theorem . with φ(t) = q t. Moreover, for each < b ≤ p,
μb (x) =
⎧ ⎨b,
if x = ;
⎩, otherwise
is a common fixed point of {Ti }∞ i= .
4 Conclusion In this work, by using the new concept of Hausdorff G-metric in the space of fuzzy sets, we establish some common fixed point theorems for a family of fuzzy self-mappings in the space of fuzzy sets on a complete G-metric space. These results are useful in fractal. An iterated function system (i.e., IFS) is the significant content in fractal, and the attractor of the IFS plays a very important role in the fractal graphics. On account of the fuzziness of parameters in fractal, by Zadeh’s extension principle [], we can get an iterated fuzzy
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function system (i.e., IFFS) corresponding the IFS []. For example, {Ti }ni= in Example . is an IFFS and {μb : < b ≤ p} ⊆ A, where A is the set of attractors of IFFS. Moreover, we can estimate the area of attractors basing on the fixed points of {Ti }ni= . Our results are also useful in fuzzy differential equation. As we all know, the existence of a solution for a fuzzy differential equation can be established via the fixed point analysis approach (see [–]). Therefore, our results provide a new method for studying the fuzzy differential equation in G-metric spaces. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the work. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, Nanchang University, Nanchang, 330031, P.R. China. 2 Department of Mathematics, Jiangxi Agricultural University, Nanchang, 330045, P.R. China. Acknowledgements The authors thank the editor and the referees for their useful comments and suggestions. The research was supported by the National Natural Science Foundation of China (11071108) and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (2010GZS0147, 20114BAB201007, 20114BAB201003). Received: 9 June 2012 Accepted: 6 September 2012 Published: 19 September 2012 References 1. Mustafa, Z, Sims, B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7, 289-297 (2006) 2. Abbas, M, Khan, AR, Nazir, T: Coupled common fixed point results in two generalized metric spaces. Appl. Math. Comput. 217, 6328-6336 (2011) 3. Abbas, M, Nazir, T, Dori´c, D: Common fixed point of mappings satisfying (E.A) property in generalized metric spaces. Appl. Math. Comput. 218, 7665-7670 (2012) 4. Abbas, M, Nazir, T, Vetro, P: Common fixed point results for three maps in G-metric spaces. Filomat 25, 1-17 (2011) 5. Choudhury, BS, Maity, P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 54, 73-79 (2011) 6. Mustafa, Z, Sims, B: Fixed point theorems for contractive mappings in complete G-metric spaces. Fixed Point Theory Appl. 2009, Article ID 917175 (2009) 7. Shatanawi, W: Fixed point theory for contractive mappings satisfying -maps in G-metric spaces. Fixed Point Theory Appl. 2010, Article ID 181650 (2010) 8. Abbas, M, Nazir, T, Radenovi´c, S: Common fixed point of generalized weakly contractive maps in partially ordered G-metric spaces. Appl. Math. Comput. 218, 9383-9395 (2012) 9. Aydi, H, Damjanovi´c, B, Samet, B, Shatanawi, W: Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces. Math. Comput. Model. 54, 2443-2450 (2011) 10. Aydi, H, Postolache, M, Shatanawi, W: Coupled fixed point results for (ψ , ϕ )-weakly contractive mappings in ordered G-metric spaces. Comput. Math. Appl. 63, 298-309 (2012) 11. Cho, YJ, Rhoades, BE, Saadati, R, Samet, B, Shatanawi, W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl. 2012, 8 (2012). doi:10.1186/1687-1812-2012-8 12. Luong, NV, Thuan, NX: Coupled fixed point theorems in partially ordered G-metric spaces. Math. Comput. Model. 55, 1601-1609 (2012) 13. Saadati, R, Vaezpour, SM, Vetro, P, Rhoades, BE: Fixed point theorems in generalized partially ordered G-metric spaces. Math. Comput. Model. 52, 797-801 (2010) 14. Kaewcharoen, A, Kaewknao, A: Common fixed points for single-valued and multi-valued mappings in G-metric spaces. Int. J. Math. Anal. 5, 1775-1790 (2011) 15. Tahat, N, Aydi, H, Karapinar, E, Shatanawi, W: Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces. Fixed Point Theory Appl. 2012, 48 (2012). doi:10.1186/1687-1812-2012-48 16. Heilpern, S: Fuzzy mappings and fuzzy fixed point theorems. J. Math. Anal. Appl. 83, 566-569 (1981) 17. Abu-Donia, HM: Common fixed point theorems for fuzzy mappings in metric space under ϕ -contraction condition. Chaos Solitons Fractals 34, 538-543 (2007) 18. Azam, A, Arshad, M, Beg, I: Fixed points of fuzzy contractive and fuzzy locally contractive maps. Chaos Solitons Fractals 42, 2836-2841 (2009) 19. Beg, I, Ahmed, MA: Common fixed point for generalized fuzzy contraction mappings satisfying an implicit relation. Appl. Math. Lett. (2012). doi:10.1016/j.aml.2011.12.019 ´ c, L, Abbas, M, Damjanovi´c, B: Common fuzzy fixed point theorems in ordered metric spaces. Math. Comput. 20. Ciri´ Model. 53, 1737-1741 (2011) 21. Kamran, T: Common fixed points theorems for fuzzy mappings. Chaos Solitons Fractals 38, 1378-1382 (2008) 22. Lee, BS, Lee, GM, Cho, SJ, Kim, DS: A common fixed point theorem for a pair of fuzzy mappings. Fuzzy Sets Syst. 98, 133-136 (1998)
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doi:10.1186/1687-1812-2012-159 Cite this article as: Zhu et al.: Common fixed point theorems for fuzzy mappings in G-metric spaces. Fixed Point Theory and Applications 2012 2012:159.
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